Sub-Nyquist sampling for short pulses with Gabor frames

For analog signals comprised of several, possibly overlapping, finite duration pulses with unknown shapes and time positions, an efficient sub-Nyquist sampling systems is based on Gabor frames. To improve the realizability of this sampling system, we present alternative method for the case that Gabor windows are high order exponential reproducing windows. Then, the time translation element could be realized with exponential filters. In this paper, we also construct the measurement matrix and prove that it has better coherence than Fourier matrix. Besides, for satisfying restricted isometry property (RIP), we reduce the row number and the sparsity by stretching windows and raising E-spline smoothness order. We deduce the signal reconstruction error bound, proving that appropriate selection of the stretching factor and smoothness order guarantees low reconstruction error. At last, we also provide the error bound in presence of noise, showing that the sampling scheme holds nice robustness with high Gabor frames redundancy.